Advertisement

Period-adding and spiral organization of the periodicity in a Hopfield neural network

  • 338 Accesses

  • 17 Citations

Abstract

This work reports two-dimensional parameter space plots, concerned with a three-dimensional Hopfield-type neural network with a hyperbolic tangent as the activation function. It shows that typical periodic structures embedded in a chaotic region, called shrimps, organize themselves in two independent ways: (i) as spirals that individually coil up toward a focal point while undergo period-adding bifurcations and, (ii) as a sequence with a well-defined law of formation, constituted by two different period-adding sequences inserted between.

This is a preview of subscription content, log in to check access.

Access options

Buy single article

Instant unlimited access to the full article PDF.

US$ 39.95

Price includes VAT for USA

Subscribe to journal

Immediate online access to all issues from 2019. Subscription will auto renew annually.

US$ 99

This is the net price. Taxes to be calculated in checkout.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  1. 1.

    Gallas JAC (1993) Structure of the parameter space of the Hénon map. Phys Rev Lett 70:2714–2717

  2. 2.

    Bonatto C, Garreau JC, Gallas JAC (2005) Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser. Phys Rev Lett 95:143905

  3. 3.

    Albuquerque HA, Rubinger RM, Rech PC (2008) Self-similar structures in a 2D parameter-space of an inductorless Chua’s circuit. Phys Lett A 372:4793–4798

  4. 4.

    Bonatto C, Gallas JAC (2008) Periodicity hub and nested spirals in the phase diagram of a simple resistive circuit. Phys Rev Lett 101:054101

  5. 5.

    Bonatto C, Gallas JAC, Ueda Y (2008) Chaotic phase similarities and recurrences in a damped-driven Duffing oscillator. Phys Rev E 77:026217

  6. 6.

    Freire JG, Bonatto C, DaCamara CC, Gallas JAC (2008) Multistability, phase diagrams, and intransitivity in the Lorenz-84 low-order atmospheric circulation model. Chaos 18:033121

  7. 7.

    Cardoso JCD, Albuquerque HA, Rubinger RM (2009) Complex periodic structures in bi-dimensional bifurcation diagrams of a RLC circuit model with a nonlinear NDC device. Phys Lett A 373:2050–2053

  8. 8.

    Freire JG, Gallas JAC (2010) Non-Shilnikov cascades of spikes and hubs in a semiconductor laser with optoelectronic feedback. Phys Rev E 82:037202

  9. 9.

    Medeiros ES, de Souza SLT, Medrano-T RO, Caldas IL (2010) Periodic window arising in the parameter space of an impact oscillator. Phys Lett A 374:2628–2635

  10. 10.

    Ramirez-Avila GM, Gallas JAC (2010) How similar is the performance of the cubic and the piecewise-linear circuits of Chua? Phys Lett A 375:143–148

  11. 11.

    Rech PC (2010) Self-similarities and period-adding in the parameter-space of a nonlinear resonant coupling process. Int J Nonlinear Sci 10:179–185

  12. 12.

    Slipantschuk J, Ullner E, Baptista MS, Zeineddine M, Thiel M (2010) Abundance of stable periodic behavior in a Red Grouse population model with delay: a consequence of homoclinicity. Chaos 20:045117

  13. 13.

    Stegemann C, Albuquerque HA, Rech PC (2010) Some two-dimensional parameter spaces of a Chua system with cubic nonlinearity. Chaos 20:023103

  14. 14.

    Testoni GE, Rech PC (2010) Dynamics of a particular Lorenz type system. Int J Mod Phys C 21:973–982

  15. 15.

    Kovanis V, Gavrielides A, Gallas JAC (2011) Labyrinth bifurcations in optically injected diode lasers. Eur Phys J D 58:181–186

  16. 16.

    Nascimento MA, Gallas JAC, Varela H (2011) Self-organized distribution of periodicity and chaos in an electrochemical oscillator. Phys Chem Chem Phys 13:441–446

  17. 17.

    Rech PC (2011) Dynamics of a neuron model in different two-dimensional parameter-spaces. Phys Lett A 375:1461–1464

  18. 18.

    Krüger TS, Rech PC (2012) Dynamics of an erbium-doped fiber dual-ring laser. Eur Phys J D 66:12

  19. 19.

    Mathias AC, Rech PC (2012) Hopfield neural network: the hyperbolic tangent and the piecewise-linear activation functions. Neural Netw 34:42–45

  20. 20.

    Maranhão DM, Baptista MS, Sartorelli JC, Caldas IL (2008) Experimental observation of a complex periodic window. Phys Rev E 77:037202

  21. 21.

    Stoop R, Benner P, Uwate Y (2010) Real-world existence and origins of the spiral organization of shrimp-shaped domains. Phys Rev Lett 105:074102

  22. 22.

    Hopfield JJ (1984) Neurons with graded response have collective computational properties like those of two-state neurons. Proc Natl Acad Sci USA 81:3088–3092

  23. 23.

    Korner E, Kupper R, Rahman MKM, Shkuro Y (2007) Neurocomputing research developments. Nova Science Publishers, New York

  24. 24.

    Zheng H, Wang H (2012) Improving pattern discovery and visualisation with self-adaptive neural networks through data transformations. Int J Mach Learn Cybern 3:173–182

  25. 25.

    Gan Q (2013) Synchronization of competitive neural networks with different time scales and time-varying delay based on delay partitioning approach. Int J Mach Learn Cybern 4:327–337

  26. 26.

    Huang W-Z, Huang Y (2011) Chaos, bifurcations and robustness of a class of Hopfield neural networks. Int J Bifurc Chaos 21:885–895

  27. 27.

    Chen P-F, Chen Z-Q, Wu W-J (2010) A novel chaotic system with one source and two saddle-foci in Hopfield neural networks. Chin Phys B 19:040509

  28. 28.

    Zheng P, Tang W, Zhang J (2010) Some novel double-scroll chaotic attractors in Hopfield networks. Neurocomputing 73:2280–2285

  29. 29.

    Wiggins S (2003) Introduction to applied nonlinear dynamical systems and Chaos. Springer, New York

  30. 30.

    Gallas JAC (2010) The structure of infinite periodic and chaotic hub cascades in phase diagrams of simple autonomous flows. Int J Bifurc Chaos 20:197–211

  31. 31.

    Albuquerque HA, Rech PC (2012) Spiral periodic structure inside chaotic region in parameter-space of a Chua circuit. Int J Circ Theor Appl 40:189–194

  32. 32.

    Vitolo R, Glendinning P, Gallas JAC (2011) Global structure of periodicity hubs in Lyapunov phase diagrams of dissipative flows. Phys Rev E 84:016216

  33. 33.

    Barrio R, Blesa F, Serrano S, Shilnikov A (2011) Global organization of spiral structures in biparameter space of dissipative systems with Shilnikov saddle-foci. Phys Rev E 84:035201(R)

  34. 34.

    Freire JG, Gallas JAC (2011) Stern–Brocot trees in the periodicity of mixed-mode oscillations. Phys Chem Chem Phys 13:12191–12198

Download references

Acknowledgments

The author thanks Conselho Nacional de Desenvolvimento Cientí fico e Tecnológico (CNPq), Brazil, for financial support.

Author information

Correspondence to Paulo C. Rech.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rech, P.C. Period-adding and spiral organization of the periodicity in a Hopfield neural network. Int. J. Mach. Learn. & Cyber. 6, 1–6 (2015). https://doi.org/10.1007/s13042-013-0222-0

Download citation

Keywords

  • Hopfield neural network
  • Hyperbolic tangent activation function
  • Lyapunov exponents
  • Period-adding bifurcation